Question

if z1,z2 are two comlex number such that |z1|=|z2|=2 and |z1+z2|=√3 then |z1-|z2|=?
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Answer 1

Step-by-step explanation:

since we have,

$|z_{1} | = |z_{2} |$

it implies that, z1 and z2 are conjugate of each other.

Let,

$|z_{1} | = x + iy \: \: and \: \: |z_{2} | = x - iy$

Given that,

$|z_{1} | = |z_{2} | = 2 \\ = > \sqrt{ {x}^{2} + {y}^{2} } = 2 \\ = > {x}^{2} + {y}^{2} = 4 \: \: \: \:$

$|z_{1} + z_{2} | = \sqrt{3} \: \: \: \: \: \: \: \: \\ = > |x + iy + x - iy| = \sqrt{3} \\ = > |2x| = \sqrt{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ = > 4 {x}^{2} = 3 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ = > {x}^{2} = \frac{3}{4} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Now,

$\: \: \: \: \: \: \: \: \: {y}^{2} = 4 - {x}^{2} \\ = > {y}^{2} = 4 - \frac{3}{4} \\ = > {y}^{2} = \frac{1}{4} \: \: \: \: \: \: \: \:$

so,

$|z_{1} - z_{2} | = |x + iy - x + iy| \\ = |2iy| = \sqrt{ {(2y)}^{2}} = \sqrt{4 {y}^{2} } = \sqrt{ 4 \times \frac{1}{4}} \\ = 1 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$